SymPy is an open source computer algebra system (CAS) written in Python.

The recent addition of the array expression module provides an alternative to the matrix expression module, with generalized support to higher dimensions (matrices are constrained to 2 dimensions).

Given the importance of multidimensional arrays in machine learning and mathematical optimization problems, this talk will illustrate examples of tensorial expressions in mathematics and how they can be manipulated using either module or in the index-explicit way.

Conversion tools have been provided to SymPy to allow users to switch an expression between the array form and either the matrix or index-explicit form. In particular, the conversion from array to matrix form attempts to represent contractions, diagonalizations and axis-permutations with operations commonly used in matrix algebra, such as matrix multiplication, transposition, trace, Hadamard and Kronecker products.

A gradient algorithm for array expressions has been implemented, returning a closed-form array expression equivalent to the derivative of arrays by arrays. The derivative algorithm for matrix expressions now uses this algorithm, attempting to convert the array back to matrix form if trivial dimensions can be dropped.